Regression Analysis

• Data analytics helps businesses make informed decisions through data interpretation
• Common data analytics techniques include regression, logistic regression, decision trees, random forests, and artificial neural networks

Regression

• Regression analysis estimates the relationship between a dependent variable and one or more independent variables
• It is used for prediction and forecasting, the aim being to estimate the value of the dependent variable based on the values of the independent variables
• Simple linear regression involves one independent variable, multiple linear regression involves several independent variables
• Simple linear regression model: y = β0 + β1x + ε, where y is the dependent variable, x is the independent variable, β0 is the intercept, β1 is the slope, and ε is the error term
• Multiple linear regression model: y = β0 + β1x1 + β2x2 + ... + βnxn + ε, where y is the dependent variable, x1, x2, ..., xn are the independent variables, β0 is the intercept, β1, β2, ..., βn are the coefficients, and ε is the error term
• Regression models are evaluated using metrics such as R-squared, Mean Squared Error (MSE), and Root Mean Squared Error (RMSE)
• R-squared measures the proportion of variance in the dependent variable that can be predicted from the independent variables
• MSE calculates the average of the squares of the differences between the predicted and actual values
• RMSE represents the square root of the MSE and provides a more interpretable measure of the prediction error
• Assumptions of linear regression include linearity, independence of errors, homoscedasticity (constant variance of errors), and normality of errors
• Violations of these assumptions can affect the reliability and validity of the regression results

Logistic Regression

• Logistic regression is used to predict the probability of a binary outcome
• The dependent variable is categorical with two possible outcomes (e.g., yes/no, true/false)
• Logistic regression uses the sigmoid function to model the relationship between the independent variables and the probability of the outcome
• The sigmoid function is defined as: p = 1 / (1 + e^(-z)), where p is the probability and z is a linear combination of the independent variables
• The coefficients are estimated to maximize the likelihood of observing the actual outcomes
• Logistic regression coefficients are interpreted as the change in the log-odds of the outcome for a one-unit change in the predictor variable
• The odds ratio is calculated by exponentiating the coefficient
• Logistic regression models are evaluated using metrics such as accuracy, precision, recall, F1-score, and AUC-ROC
• Accuracy measures the proportion of correct predictions
• Precision measures the proportion of positive predictions that are actually correct
• Recall measures the proportion of actual positive cases that are correctly predicted
• F1-score represents the harmonic mean of precision and recall
• AUC-ROC measures the ability of the model to discriminate between the two classes

Decision Tree

• Decision trees constitute a non-parametric supervised learning method used for both classification and regression tasks
• A decision tree uses a tree-like structure to model the relationship between the features and the target variable
• The tree is constructed by recursively splitting the data based on the values of the features
• The splitting criterion aims to maximize the separation of the target variable
• Common splitting criteria include Gini impurity, entropy, and information gain for classification trees
• Variance reduction and mean squared error represent common splitting criteria for regression trees
• Decision trees are easy to interpret and visualize, which makes them useful for understanding the relationships between variables
• Decision trees can be prone to overfitting, especially if the tree is very deep
• Techniques for preventing overfitting include limiting the depth of the tree, setting a minimum number of samples required to split a node, and pruning the tree

Random Forest

• Random forest is an ensemble learning method that combines multiple decision trees to improve predictive performance
• It builds multiple decision trees on random subsets of the data and random subsets of the features
• Random forest reduces the risk of overfitting and improves the generalization ability of the model
• The ultimate prediction is made by averaging the predictions of all the individual trees
• Random forests can be used for both classification and regression tasks
• Random forests provide estimates of feature importance, which can be used to identify the most relevant predictors
• Random forests are relatively robust to outliers and missing values
• Tuning parameters for random forests include the number of trees, the maximum depth of the trees, and the number of features to consider at each split

Artificial Neural Network

• Artificial Neural Networks (ANNs) are machine learning models inspired by the structure and function of the human brain
• ANNs consist of interconnected nodes (neurons) organized in layers
• The basic structure includes an input layer, one or more hidden layers, and an output layer
• Each connection between neurons has a weight associated with it, representing the strength of the connection
• Neurons apply an activation function to the weighted sum of their inputs to produce an output
• Common activation functions include sigmoid, ReLU (Rectified Linear Unit), and tanh (hyperbolic tangent)
• ANNs learn through a process called training, where the weights are adjusted to minimize the difference between the predicted and actual outputs
• Training algorithms such as backpropagation are used to update the weights based on the error gradient
• ANNs can learn complex non-linear relationships between variables
• ANNs require large amounts of data to train effectively and are computationally intensive
• They are used in a wide range of applications, including image recognition, natural language processing, and time series forecasting
• Hyperparameters such as the number of layers, the number of neurons per layer, the learning rate, and the batch size need to be tuned to achieve optimal performance
• Overfitting is a common problem in ANNs, and techniques such as regularization, dropout, and early stopping are used to prevent it